A052980 -id:A052980 - cp's OEIS frontend

A277733 Positions of 1's in A277731.

2, 4, 7, 9, 13, 15, 18, 20, 24, 26, 28, 31, 33, 37, 39, 42, 44, 48, 50, 52, 55, 57, 60, 62, 66, 68, 71, 73, 77, 79, 81, 84, 86, 90, 92, 95, 97, 101, 103, 105, 108, 110, 113, 115, 119, 121, 124, 126, 130, 132, 135, 137, 141, 143, 145, 148, 150, 154, 156, 159, 161, 165, 167, 169, 172, 174, 177, 179

Offset: 1

N. J. A. Sloane, Nov 07 2016

nonn

{A277732, A277733, A277734} forms a three-way partition of the positive integers, similar to {A003144, A003145, A003146}.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A052980, A277731, A277732, A277734.

Cf. also A003144, A003145, A003146.

Maple
```
See A277732.
```

m = 1000; (* number of terms of A277731 *)
S[1] = {0};
S[n_] := S[n] = SubstitutionSystem[{0 -> {0, 1}, 1 -> {0, 1, 2}, 2 -> {0}}, S[n - 1]];
For[n = 2, True, n++, If[PadRight[S[n], m] == PadRight[S[n - 1], m], Print["n = ", n]; Break[]]];
A277731 = Take[S[n], m];
Position[A277731, 1] // Flatten (* Jean-François Alcover, Mar 20 2023 *)

A277734 Positions of 2's in A277731.

Original entry on oeis.org

5, 10, 16, 21, 29, 34, 40, 45, 53, 58, 63, 69, 74, 82, 87, 93, 98, 106, 111, 116, 122, 127, 133, 138, 146, 151, 157, 162, 170, 175, 180, 186, 191, 199, 204, 210, 215, 223, 228, 233, 239, 244, 250, 255, 263, 268, 274, 279, 287, 292, 298, 303, 311, 316, 321, 327, 332, 340, 345, 351, 356, 364, 369, 374

Offset: 1

N. J. A. Sloane, Nov 07 2016

nonn

{A277732, A277733, A277734} forms a three-way partition of the positive integers, similar to {A003144, A003145, A003146}.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A052980, A277731, A277732, A277733.

Cf. also A003144, A003145, A003146.

Maple
```
See A277732.
```

m = 1000; (* number of terms of A277731 *)
S[1] = {0};
S[n_] := S[n] = SubstitutionSystem[{0 -> {0, 1}, 1 -> {0, 1, 2}, 2 -> {0}}, S[n - 1]];
For[n = 2, True, n++, If[PadRight[S[n], m] == PadRight[S[n - 1], m], Print["n = ", n]; Break[]]];
A277731 = Take[S[n], m];
Position[A277731, 2] // Flatten (* Jean-François Alcover, Mar 20 2023 *)

A114188 Riordan array (1/(1-x),x(1+x)/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 7, 1, 1, 16, 26, 10, 1, 1, 25, 70, 52, 13, 1, 1, 36, 155, 190, 87, 16, 1, 1, 49, 301, 553, 403, 131, 19, 1, 1, 64, 532, 1372, 1462, 736, 184, 22, 1, 1, 81, 876, 3024, 4446, 3206, 1216, 246, 25, 1, 1, 100, 1365, 6084, 11826, 11584, 6190, 1870, 317

Offset: 0

Paul Barry, Nov 16 2005

Product of A007318 and A113953, that is, (1/(1-x),x/(1-x))*(1,x(1+2x)).
Row sums are A025192. Diagonal sums are A052980.
Inverse is A114189. A signed version is A110511.

			Triangle begins
1;
1, 1;
1, 4, 1;
1, 9, 7, 1;
1, 16, 26, 10, 1;
1, 25, 70, 52, 13, 1;
1, 36,155,190, 87, 16, 1;

P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

Cf. A007318, A025192, A052980, A110511, A113953, A114189.

T(n, k) = Sum_{j=0..n} C(n, j)*C(k, j-k)2^(j-k).
T(n, k) = Sum_{j=0..n-k} C(k, j)*C(n+k-j, 2k).
T(n,k) = 2*T(n-1,k)+T(n-1,k-1)-T(n-2,k)+T(n-2,k-1), T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 11 2014
G.f.: 1/(1-y-x*(1+y)/(1-y)). - Vladimir Kruchinin, Apr 21 2015

A329185 Number of ways to tile a 2 X n grid with dominoes and L-trominoes such that no four tiles meet at a corner.

Original entry on oeis.org

1, 1, 2, 5, 10, 22, 49, 105, 227, 494, 1071, 2322, 5038, 10927, 23699, 51405, 111498, 241837, 524546, 1137742, 2467761, 5352577, 11609747, 25181550, 54618807, 118468250, 256957750, 557341615, 1208874523, 2622050045, 5687229162, 12335605733, 26755941146

Offset: 0

Peter Kagey, Nov 07 2019

a(n) <= A052980(n).

			For n=3, the five tilings are:
+---+---+---+  +---+---+---+
|   |   |   |  |   |       |
+   +   +   +  +   +---+---+
|   |   |   |  |   |       |
+---+---+---+, +---+---+---+,
+---+---+---+  +---+---+---+
|       |   |  |   |       |
+---+---+   +  +   +---+   +
|       |   |  |       |   |
+---+---+---+, +---+---+---+, and
+---+---+---+
|       |   |
+   +---+   +
|   |       |
+---+---+---+.
For n=4, the only tiling counted by A052980(4) that is not counted by a(4) is
+---+---+---+---+
|       |       |
+---+---+---+---+
|       |       |
+---+---+---+---+.

Peter Kagey, Table of n, a(n) for n = 0..2500
Misha Lavrov, Number of ways to tile a room with I-Shaped and L-Shaped Tiles, Mathematics Stack Exchange.
Index entries for linear recurrences with constant coefficients, signature (2,-1,3,-1,2).

A052980 is the analogous problem without the "four corners" restriction.

LinearRecurrence[{2, -1, 3, -1, 2}, {1, 1, 2, 5, 10}, 50] (* Paolo Xausa, Apr 08 2024 *)

Vec((1 - x)*(1 + x^2) / (1 - 2*x + x^2 - 3*x^3 + x^4 - 2*x^5) + O(x^30)) \\ Colin Barker, Nov 12 2019

a(n) = 2*a(n-1) - a(n-2) + 3*a(n-3) - a(n-4) + 2*a(n-5), with a(0) = a(1) = 1, a(2) = 2, a(3) = 5, and a(4) = 10.

G.f.: (1 - x)*(1 + x^2) / (1 - 2*x + x^2 - 3*x^3 + x^4 - 2*x^5). - Colin Barker, Nov 12 2019

A332647 a(n) = 2*a(n-1) + a(n-3) with a(0) = 3, a(1) = 2, a(2) = 4.

Original entry on oeis.org

3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008, 13251, 29226, 64460, 142171, 313568, 691596, 1525363, 3364294, 7420184, 16365731, 36095756, 79611696, 175589123, 387274002, 854159700, 1883908523, 4155091048, 9164341796, 20212592115, 44580275278, 98324892352

Offset: 0

Greg Dresden, Feb 18 2020

a(n) is the number of ways to tile a bracelet of length n with black trominos, and black or white squares.

Colin Barker, Table of n, a(n) for n = 0..1000
Greg Dresden and Michael Tulskikh, Tilings of 2 X n boards with dominos and L-shaped trominos, Washington & Lee University (2021).
Helmut Prodinger, On third-order Pell polynomials, arXiv:2011.04388 [math.NT], 2020.
Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Cf. A008998, A052980. Equals one more than A080204.

a:=[3,2,4]; [n le 3 select a[n] else 2*Self(n-1)+Self(n-3):n in [1..33]]; // Marius A. Burtea, Feb 18 2020

LinearRecurrence[{2, 0, 1}, {3, 2, 4}, 50]

Vec((3 - 4*x) / (1 - 2*x - x^3) + O(x^30)) \\ Colin Barker, Feb 18 2020

polsym(x^3-2*x^2-1, 44) \\ Joerg Arndt, May 28 2020

a(n) = 2*a(n-1) + a(n-3).
a(n) = w1^n + w2^n + w3^n where w1,w2,w3 are the three roots of x^3-2x^2-1=0.
For n>2, a(n) = round(w1^n) for w1 the single real root of x^3-2x^2-1=0.
G.f.: (3 - 4*x) / (1 - 2*x - x^3). - Colin Barker, Feb 18 2020
a(n) = A008998(n) + 2*A008998(n-3) = 3*A008998(n) - 4*A008998(n-1).
a(n) = (5*b(n) - b(n-1) - b(n-2))/2 where b(n) = A052980(n). - Greg Dresden, Mar 10 2020
a(n) = A080204(n) + 1. - Greg Dresden, May 27 2020

A346054 Number of ways to tile a 3 X n strip with dominoes and L-shaped 5-minoes.

Original entry on oeis.org

1, 0, 3, 8, 13, 52, 119, 308, 873, 2184, 5867, 15552, 40581, 107836, 283871, 748076, 1976545, 5208784, 13743315, 36260088, 95627773, 252289476, 665499975, 1755466916, 4630903129, 12215645848, 32223689915, 85003275440, 224228961909, 591494654412, 1560303157679

Offset: 0

Greg Dresden and Ziyao Geng, Jul 02 2021

			Here are two such tilings for a 3 X 3 strip; each has four rotations thus demonstrating that a(3)=8.
  ._____.  ._____.
  | | | |  | |___|
  | |_|_|  | |___|
  |_____|  |_____|
For a 3 X 4 strip, here are three of the possible a(4)=13 tilings.
  ._______.  ._______.  ._______.
  | |___  |  |  ___| |  |___|___|
  | |___| |  | |___| |  | |___| |
  |_____|_|  |_|_____|  |_|___|_|
For a 3 X 5 strip, here are three of the possible a(5)=52 tilings.
  ._________.  ._________.  ._________.
  | | |___| |  |  ___|___|  | |___|___|
  | |_|___|_|  | | |___| |  | |___|___|
  |_____|___|  |_|_|___|_|  |_____|___|

G. C. Greubel, Table of n, a(n) for n = 0..1000
Greg Dresden and Michael Tulskikh, Tilings of 2 X n boards with dominos and L-shaped trominos, Journal of Integer Sequences 24 (2021), article 21.4.5.
Index entries for linear recurrences with constant coefficients, signature (1,3,5,-4).

Cf. A052980.

I:=[1,0,3,8]; [n le 4 select I[n] else Self(n-1) +3*Self(n-2) +5*Self(n-3) -4*Self(n-4): n in [1..50]]; // G. C. Greubel, Dec 01 2022

LinearRecurrence[{1, 3, 5, -4}, {1, 0, 3, 8}, 50];

@CachedFunction
def a(n): # a = A346054
    if (n<4): return (1,0,3,8)[n]
    else: return a(n-1) + 3*a(n-2) + 5*a(n-3) - 4*a(n-4)
[a(n) for n in range(51)] # G. C. Greubel, Dec 01 2022

a(n) = a(n-1) + 3*a(n-2) + 5*a(n-3) - 4*a(n-4).

G.f.: (1 - x)/(1 - x - 3*x^2 - 5*x^3 + 4*x^4).

Corrected by Greg Dresden, Sep 04 2021

A180752 Half the number of nX3 binary arrays with each element equal to at least two neighbors.

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 14, 31, 67, 148, 325, 717, 1580, 3485, 7685, 16950, 37383, 82451, 181850, 401083, 884615, 1951080, 4303241, 9491097, 20933272, 46169785, 101830665, 224594602, 495358987, 1092548639, 2409691878, 5314742743

Offset: 1

R. H. Hardin Sep 20 2010

nonn

Column 3 of A180760

R. H. Hardin, Table of n, a(n) for n=1..99

Empirical: a(n) = 2*a(n-1) + a(n-2) - a(n-3) - a(n-5).

Empirical: G.f.: -x^2*(-1+x+x^3+x^2) / ( (x-1)*(1+x)*(x^3+2*x-1) ). a(n)-a(n-2) = A052980(n-3). - R. J. Mathar, Mar 24 2018

A190512 Number of one-sided n-step prudent walks, avoiding single west step only, i.e., two or more consecutive west steps are permitted.

Original entry on oeis.org

1, 2, 5, 11, 24, 53, 117, 258, 569, 1255, 2768, 6105, 13465, 29698, 65501, 144467, 318632, 702765, 1549997, 3418626, 7540017, 16630031, 36678688, 80897393, 178424817, 393528322, 867954037, 1914332891, 4222194104, 9312342245, 20539017381, 45300228866

Offset: 0

Shanzhen Gao, May 11 2011

			a(2)=5 since there are 5 such walks: WW, NN, EN, NE, EE.

S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.
Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Cf. A110513 (essentially a signed version).

Cf. A052980 (essentially the same sequence).

my(x='x+O('x^35)); Vec((1+x^2)/(1-2*x-x^3)) \\ Michel Marcus, Jun 28 2021

a(n) = A052980(n+1). - R. J. Mathar, May 16 2011

G.f.: (1+x^2)/(1-2*x-x^3).

A335242 a(n) = 2*a(n-1) + a(n-3) for n >= 4, with initial values a(0) = 1, a(1) = 0, a(2) = 2, and a(3) = 3.

Original entry on oeis.org

1, 0, 2, 3, 6, 14, 31, 68, 150, 331, 730, 1610, 3551, 7832, 17274, 38099, 84030, 185334, 408767, 901564, 1988462, 4385691, 9672946, 21334354, 47054399, 103781744, 228897842, 504850083, 1113481910, 2455861662, 5416573407, 11946628724, 26349119110, 58114811627

Offset: 0

Greg Dresden, May 28 2020

a(n) is the number of ways to tile this 2 X n strip (with one extra square added at the top left) with dominoes and L-shaped trominoes (also called polyominoes):
._
|| _  
|||_||| . . .
|||_||| . . .

			a(2) = 2 thanks to the following two tilings (where the L-shaped trominoes are tiled with X's and the dominoes are left blank):
._            _
|X|_         | |_
|X|X|  and   |_|X|
|_ _|        |X X|

Index entries for linear recurrences with constant coefficients, signature (2,0,1).

Cf. A052980, A008998.

LinearRecurrence[{2, 0, 1}, {1, 0, 2, 3}, 40]

a(n) = 2*a(n-1) + a(n-3) for n >= 4.
a(n) = A008998(n-2) + A052980(n-2) for n >= 2.
G.f.: (2*x^3-2*x^2+2*x-1)/(x^3+2*x-1).

A100691 Number of self-avoiding paths with n steps on a triangular lattice in the strip Z x {0,1}.

Original entry on oeis.org

1, 4, 12, 30, 70, 158, 352, 780, 1724, 3806, 8398, 18526, 40864, 90132, 198796, 438462, 967062, 2132926, 4704320, 10375708, 22884348, 50473022, 111321758, 245527870, 541528768, 1194379300, 2634286476, 5810101726, 12814582758

Offset: 0

Emeric Deutsch, Dec 07 2004

J. Labelle, Paths in the Cartesian, triangular and hexagonal lattices, Bulletin of the ICA, 17, 1996, 47-61.

Index entries for linear recurrences with constant coefficients, signature (3,-2,1,-1).

g:=series((1+z^2)*(1+z+z^2)/(1-z)/(1-2*z-z^3),z=0,35): 1,seq(coeff(g,z^n), n=1..34);

G.f.: (1+z^2)(1+z+z^2)/[(1-z)(1-2z-z^3)]= 1+2*(2+z^2)/((z-1)*(z^2+2*z-1)).
a(n) = 2*a(n-1) + a(n-3) + 6 for n >= 4.
a(n) = A008998(n+2) - A052980(n+1) - 3. - Ralf Stephan, May 15 2007
Conjecture: a(n) = A193641(n+2)-3, n>0 - R. J. Mathar, Jul 22 2022

cp's OEIS Frontend

A277733 Positions of 1's in A277731.

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A277734 Positions of 2's in A277731.

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A114188 Riordan array (1/(1-x),x(1+x)/(1-x)^2).

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A329185 Number of ways to tile a 2 X n grid with dominoes and L-trominoes such that no four tiles meet at a corner.

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A332647 a(n) = 2*a(n-1) + a(n-3) with a(0) = 3, a(1) = 2, a(2) = 4.

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A346054 Number of ways to tile a 3 X n strip with dominoes and L-shaped 5-minoes.

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A180752 Half the number of nX3 binary arrays with each element equal to at least two neighbors.

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A190512 Number of one-sided n-step prudent walks, avoiding single west step only, i.e., two or more consecutive west steps are permitted.

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